Long-term evolution of Landau damping in the presence of transverse noise, feedback, and detuning

The effect of Landau damping is often calculated assuming a Gaussian beam distribution in all transverse degrees of freedom, which agrees reasonably well with beam measurements. However, the stability of the beam is strongly dependent on the details of the distribution. The present study focuses on the slow evolution of the transverse bunch distribution, for a bunch excited by a coherent white noise source, damped by a transverse feedback system, and with detuning dependent on the transverse actions. The mechanism is modeled by the Fokker-Planck equation. It corresponds to a diffusion that is zero for particles with tune equal to the average tune of the bunch, and which is growing quadratically with the tune in the vicinity. The evolving distributions are then used to calculate the evolving stability diagrams, and thus the long-term evolution of the Landau damping. The relative effective octupole current is reduced faster for stronger noise, stronger linear detuning coefficients and weaker damper gain. The relevant parameters for this mechanism arc scanned. With relevant parameter values for the LHC, this mechanism can cause a reduction of the effective octupole current by at most 10%/h.